Object displacement over LOD
The simple answer is that modifiers aren't fully supported by the Game Engine.

While modifiers will typically be rendered by the Game Engine, they won't be updated by the engine. So what's happening is the displace modifier is being altered by the Game Engine at the point where it needs to change LOD, but because modifiers don't update correctly, it's just reverting to the flat version without displacement.

This answer has some additional information: Make modifiers update in the game engine

You're not out of luck, though. There are other ways to implement LOD in the BGE.